Method and apparatus for transmitting high speed data by spectral decomposition of the signaling space

ABSTRACT

A method and apparatus of high speed multi-dimensional signaling via a modem has a processing method of utilizing prolate pulses to optimize the transmission capacity of the channel. The modem includes a process that segments the channel bandwidth and allocates the power and bit loading in relation to a measure of the noise in each spectral bin. Data are carried over a plurality of frequencies across the channel, and within each spectral bin, a plurality of orthogonal signaling dimensions.

CROSS REFERENCE TO RELATED APPLICATION

This application is a continuation of U.S. patent application Ser. No.09/931,782 filed Aug. 17, 2001 (now issued as U.S. Pat. No. 7,012,953),entitled, “METHOD AND APPARATUS FOR TRANSMITTING HIGH SPEED DATA BYSPECTRAL DECOMPOSITION OF THE SIGNALING SPACE”, which is a continuationin part of U.S. patent application Ser. No. 09/021,137 filed Feb. 10,1998 (issued Jun. 4, 2002 as U.S. Pat. No. 6,400,776), entitled, “METHODAND APPARATUS FOR HIGH SPEED DATA TRANSMISSION BY SPECTRAL DECOMPOSITIONOF THE SIGNALING SPACE”. Each of the aforementioned applications isincorporated herein by reference.

FIELD OF THE INVENTION

This invention is generally directed to high-speed data communication,and more specifically, to the area of high-speed modem design. Itrelates to achieving high spectral efficiency in signaling systems.

BACKGROUND OF THE INVENTION

Modern telecommunication applications have resulted in substantialincreases in the need for additional bandwidth. For example, in the areaof wired communications, there is a need to simultaneously supportvoice, video, and data applications at low BER (Bit Error Rates) usingnew high-speed modem designs for twisted pairs. At signaling ratesbetter than 10 Mbits/s performance bounds generally exceed a BER of10⁻⁶. When the conventional Pulse Amplitude Modulation (PAM) techniqueis used, the baseband communication signal is represented by a series ofmodulated pulses whose amplitude levels are determined by the symbol tobe transmitted. For example, with 16-QAM (Quadrature AmplitudeModulation), typical symbol amplitudes of ±1 and ±3 are utilized in eachquadrature channel. For digital communication systems, efficient used ofbandwidth is crucial when dealing with time-dispersive channel, as iscommon with wireless systems. In these types of systems, whenever thereis distortion of the signals due to preceding or following pulses,normally referred to as pre-cursors and post-cursors, respectively, theamplitude of the desired pulse is affected due to superimposition of theoverlapping pulses. This phenomenon is known as intersymbolinterference, and is an impediment to high-speed data transmission,especially in systems that are constrained by limited bandwidth.

One way to minimize the effects of intersymbol interference is to use anequalizer. Fixed equalizers are designed to be effectively operatedbetween an upper and lower bound between which the channel is expectedto deviate. Whenever these limits are exceeded, the equalizer ceases tooperate effectively. Hence, there has to be a certain degree ofprecision when channel equalization is employed, and fixed equalizersare implemented. There are adaptive equalizers (i.e., continuous) thattrack dynamic channel dispersion and make continuous adjustments tocompensate for such intersymbol interference. This provides someimprovement in performance over the fixed equalizer.

Incorporation of the equalizer into some communication systems does notcome without penalty. In wireless systems, for instance, insertion lossbecomes a critical factor if the equalizer is present and the associatedimpairment does not occur. The main purpose of the equalizerimplementation is to enhance the information bearing capability of thecommunication system with the design objective of asymptoticallyapproaching the capacity bounds of the transmission channel.Consequently, the use of the equalizer can be regarded as one instanceof an array of possibilities that may be implemented to enhance the bitrate of the communication system design.

SUMMARY OF THE INVENTION

In accord with the invention, a method and apparatus is provided thatmakes more efficient use of the available signaling bandwidth in thesense of asymptotically approaching possible transmission limits. Thisis done by significantly reducing the effects of intersymbol andinterchannel interference by a judicious choice of the signaling pulseshapes. In particular, prolate pulses are used to extend channelcapacity and reduce interference. By use of orthogonal axes that spanthe signal space, combined with water filling techniques for efficientallocation of transmission energy based on the noise distribution, theinformation content can be increased without increase in bandwidth.

The signaling space is spectrally decomposed to support the simultaneoustransmission of multiple signals each with differing information bearingcontent. These signals being orthogonal, are non-interfering. Signalsare constructed as complex sets and are generally represented with axialcoordinates, all orthogonal to one another within the complex plane. Thereal axis is termed the in-phase (I) component and the imaginary axis istermed the quadrature (Q) component. Each component defines a spanningvector in the signal space.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a graph depicting the concentration of energy in a prolatepulse interval;

FIG. 2 is a block schematic illustrating an optimized modem using theinvention;

FIG. 3 is a block schematic of a Single Segment Discrete ProlateTransmitter;

FIG. 4 is a graph illustrating the application of water filling to thepresent invention.

FIG. 5 is a graph depicting channel segmentation and use of thefrequency response; and

FIG. 6 is a block schematic of the Discrete Prolate demodulatorcorresponding to FIG. 3.

DETAILED DESCRIPTION

Spectral efficiency in digital systems is largely a function of the waveshapes of the signals that are used to carry the digital information.There are tradeoffs between time limitations and frequency limitations.These two requirements generally have a flexible relationship. Thecharacteristics of prolate pulses may be chosen to limit spectral energydispersion thereby permitting more signaling channels for a givenbandwidth. These advantages become readily apparent with an analysis ofthe prolate pulse spectral performance. In particular, the Fouriertransform of the waveform is very band limited. Proper selection ofsignal space such as axes or spectral vectors representing signalcoordinates are very important. If signals are orthogonal to oneanother, transmission techniques utilizing methods of water filling maybe implemented with significant increase in efficiency. The technique ofwater filling is discussed in “Multicarrier Modulation for DataTransmission: An Idea Whose Time Has Come”, J. A. C. Bingham, “IEEECommunications,” May 1990, pp. 5-14 incorporated by reference. Anillustrated of water filling may be ascertained from the graph of FIG.4. A bandwidth of a channel is defined by the marks 401 and 402 on thehorizontal axis. The curve 403 defines the noise level produced as seenby a receiver. The energy level, which the channel can transmit, isdefined by the horizontal level 404. The area 405 bounded by the curve403 and energy level 404 may be “water filled” by data signals. The dataacceptance area 405 of the band is divided into sections 408 by verticaldividers 409. The signal data is inserted into a section until the addeddata and noise in that section reaches the energy level limit. Thisfilling combined with the orthogonal nature of the data signals insertedin the sections permit the increase in the data capacity of the channel.

Consider a trigonometric polynomial p_(i)(t) defined as follows:

$\begin{matrix}{{p_{i}(t)} = {\underset{n = {- N}}{\sum\limits^{N}}{a_{in}e^{{jn}\;\pi\; t}}}} & (1)\end{matrix}$

In equation (1) the period may be chosen to be 2 by suitable scaling oft. The coefficients a_(in) can be obtained by an optimization process,the objective of which is to obtain a spectrally efficient pulse. Theprocess may be regarded as a scheme in which the energy of the pulse isconcentrated in the interval [−ε,ε]. This is shown in FIG. 1 where amore or less generic pulse 101 is shown and the constraining interval102 is indicated. The optimization process is a transmission pulsedesign problem, and a particular mathematical approach for achievingthis objective is now described. In general, optimal communicationsystem design requirements often necessitate the transmission ofspectrally efficient pulses in order to minimize both intersymbolinterference and interchannel interference where application requiressegmented spectrum utilization.

Based on the specified format in equation (1), it can be shown that thecoefficients a_(in) of p_(i)(t) satisfy the following system ofequations:

$\begin{matrix}{{{\underset{m = {- N}}{\sum\limits^{N}}{\frac{{\sin\left( {n - m} \right)}\pi\; ɛ}{\left( {n - m} \right)ɛ}a_{im}}} = {\lambda\; a_{in}}},{n = {- N}},{{- N} + 1},{\ldots\mspace{14mu}{N.}}} & (2)\end{matrix}$

Equation (2) may be rewritten in the form,Sā _(i)=λ_(i) ā _(i)  (3)

Where the coefficients S_(nm) of the matrix S defined by equation (3),and eigenvectors ā_(i) are given by,

$\begin{matrix}{S_{nm} = \frac{{\sin\left( {n - m} \right)}\pi\; ɛ}{\left( {n - m} \right)ɛ}} & (4)\end{matrix}$andā _(i) =[a _(−Ni) , a _((−N+1)i) , . . . a _(0i) , . . . , a _((N−1)i) ,a _(Ni)]^(t)  (5)

Where t denotes transpose. The matrix S is a real, symmetric, andpositive definite with other mathematical properties of interest to thedevelopment, as now discussed. There are thus 2N+1 real eigenvaluesλ_(i) which satisfy (3) and which may be ordered such that:λ₁>λ₂> . . . >λ_(2N+1)  (6)

For each eigenvalue λ_(i), there is an associated eigenvector ā_(i),whose coefficients can be used to form the trigonometric functiondefined in equation (1). The eigenvectors of the matrix S may benormalized to have unit energy. Because of the orthogonality of theeigenvectors of symmetric matrices, their dot products ā_(i)•ā_(j)satisfy the following relationship,

$\begin{matrix}{{{{\overset{\rightharpoonup}{a}}_{i} \cdot {\overset{\rightharpoonup}{a}}_{j}} = {{\underset{n = {- N}}{\sum\limits^{N}}{a_{in}a_{jn}}} = \delta_{ij}}},} & (7)\end{matrix}$

Where δ_(ij) is the Kronecker delta function. Because of equation (3)and equation (7), it can be shown that functions of the form of equation(1) whose coefficients are those of the eigenvectors of the matrix S asdefined in equation (4), the following relationships holds:

$\begin{matrix}{{{\frac{1}{2}{\int_{- 1}^{1}{{p_{i}(t)}{p_{j}(t)}{\mathbb{d}t}}}} = \delta_{ij}},} & (8)\end{matrix}$and,

$\begin{matrix}{{\frac{1}{2}{\int_{- ɛ}^{ɛ}{{p_{i}(t)}{p_{j}(t)}{\mathbb{d}t}}}} = {\lambda_{i}\delta_{ij}}} & (9)\end{matrix}$Functions so formed are described as discrete prolate.

With the background material discussed above, a particular method ofcommunicating digital information using the functions p_(i)(t) definedearlier is now presented. Again, in view of equation (6), there are 2N+1eigenvectors that satisfy equation (3). The vectors together form aspanning set for the vector space defined by the matrix S. Define D tobe the dimension of the associated vector space. Then D is given by:D=2N+1  (10)

Note that D is a design parameter, and is a function of N. By analogy,{p_(i)(t)} form a spanning set for the signal space associated with thematrix S, and this signal space is also D dimensional. Consider theconstruct:

$\begin{matrix}{{x_{i}(t)} = {\underset{k = {- \infty}}{\sum\limits^{\infty}}{I_{k}{p_{i}\left( {t - {kT}} \right)}}}} & (11)\end{matrix}$

Generalizing and using equation (8), it can be shown that the followingholds:

$\begin{matrix}{{\frac{1}{2T}{\int_{{- T} + {kT}}^{T + {kT}}{{x_{i}(t)}{p_{j}(t)}{\mathbb{d}t}}}} = {\delta_{ij}I_{k}}} & (12)\end{matrix}$

Equation (12) is of critical importance to the invention. Theimplications are that if a function of the form of equation (11), for aspecific value of i, is transmitted over a communication channel, thenthe transmitted alphabet I_(k) will only be uniquely determined in aninterval defined by k if the corresponding p_(i)(t) is used as thereceiving filter. If a function of the form of equation (11), for aspecific value of i, is transmitted over a communication channel, andp_(j)(t) for j≠i, is used as the receiving filter, then the functionp_(i)(t) will be virtually non-existent. Thus in order to extract theinformation content of a signal whose format is given by equation (11),the signaling pulse must be matched at the receiver. In anticipation ofmaking reference to Cartesian space, the format of equation (11) is usedin the construction of y_(i)(t) defined as follows:

$\begin{matrix}{{y_{i}(t)} = {\underset{k = {- \infty}}{\sum\limits^{\infty}}{Q_{k}{p_{i}\left( {t - {kT}} \right)}}}} & (13)\end{matrix}$where again Q_(k) is the alphabet to be transmitted. It is clear thatequation (13) also satisfies a relationship similar to equation (12).Equations (11) and (13) can now be used to quadrature modulate a carrierin the final part of the transmission signal synthesis. Define s_(i)(t)by:s _(i)(t)=x _(i)(t)cos(2πƒ_(c) t) −y _(i)(t)sin(2πƒ_(c) t)  (14)

Thus, the signals are constructed as complex sets and are generallyrepresented as vectors within the complex plane. The real axis is termedthe in-phase (I) component and the imaginary axis is termed thequadrature (Q) component.

As indicated by equation (10), there are D such constructs possible.Because of the orthogonality of the building blocks {p_(i)(t)} discussedearlier, {s_(i)(t)}, being linear combinations of a single p_(i)(t) foreach i, are themselves orthogonal, forming a spanning set for the signalspace defined over the channel band limited by W=1/2T. That is to say,each such signal s_(i)(t) may be regarded as an orthogonal “finger” overwhich the symbols {I_(k), Q_(k)} may be independently transmitted. Thus,equation (14) can be used to increase the bit rate of the communicationchannel without bandwidth expansion. Of course coding and equalizationmay be added to improve fidelity.

The parameters ε and N determine the spectral shape of the transmissionpulses p_(i)(t). In general ε will be used to determine the compactnessof the fit within the signaling period, while N determines the peakingand roll-off. It is important for N to be fairly large (N≧10) as thereare at least two benefits to be gained in this regard. Firstly, largevalues of N contribute to better roll-off characteristics, whichdirectly minimize intersymbol interference. Secondly, as can be seen byequation (10), large values of N contribute directly to an increase inthe dimension of the signaling space, providing more discrete prolatefunctions that can be used to increase the capacity of the transmissionsystem design. However, these benefits must be balanced by the fact thattighter peaks that are made possible by larger values of N are likely toplace greater implementation constraints on the receiver, to the extentthat more accurate symbol timing shall be required to retrieve theencoded digital information.

In general, the range of values that can be taken on by the discretesymbols {I_(k), Q_(k)} determines the number of levels M that may bereasonably distinguished at the receiver, with noise, crosstalk, andinterference playing a critical role in the process. Conventionalmodulation techniques, such as QAM for instance, may be referenced, andthe value of M shall be determined in an optimization process in whichthe power is held constant, and the bit rate is maximized for a givenBER constraint. Given M, the information bearing capacity C of thetransmitter is computed in a straightforward manner. Thus,

$\begin{matrix}{C = \frac{\log_{2}M}{T}} & (15)\end{matrix}$

Where C is given in units of bits/s. Equation (15) holds for theone-dimensional case. That is, only when one signal of duration T havingM reasonably distinguishable levels is transmitted in a channelbandlimited by W. However, when multiple orthogonal signaling is usedfor data transmission, the parameter M in equation (15) will be afunction of the number of signals chosen, along with the associatedlevels that may be represented by each independent choice. A limitingreformulation of equation (15) is now given by:

$\begin{matrix}{C_{\lim} = \frac{\log_{2}{\prod\limits_{i}M_{i}}}{T}} & (16)\end{matrix}$

Equation (16) hints of the possibility of channel optimization with moreefficient encoding of the signaling data. A concrete use of equation(16) is demonstrated in the sequel, specifically, with the aid of FIGS.3 and 6.

The area of optimal communication system design is generally one inwhich various signal-processing techniques are comprised toasymptotically approach theoretically established channel capacitylimits. Transmission rates may further be optimized if a process knownas water filling is implemented. With the implementation of waterfilling, the available signaling power is allocated to the communicationchannel, and the bits are loaded in a manner related to the noisespectral density, with the objective of maximizing resource utility. Itmay be regarded as a process in which the sqander of the availablesignal energy is avoided. Let the noise be Gaussian, with the powerspectral density given by N(ƒ), with H(ƒ) being the associated complextransfer function of the channel. Then, in order to make efficient useof the available signaling power S, the optimal channel input power isgiven by:

$\begin{matrix}{S = {{\int_{f \in \Omega}B} - {\frac{N(f)}{{{H(f)}}^{2}}{\mathbb{d}f}}}} & (17)\end{matrix}$where the region of integration Ω is defined by:

$\begin{matrix}{\Omega = \left\{ {f:{\frac{N(f)}{{{H(f)}}^{2}} \leq B}} \right\}} & (18)\end{matrix}$

Equations (17) and (18) describe the elastic relationship that existsbetween the power spectral densities of the input signal and the noiseduring the process of optimizing the available channel bandwidth. Inequations (17) and (18) B is an average input power constraint.

From a practical standpoint, the optimal allocation of signaling poweris best achieved by channel segmentation. Then, the allocation of bitsto the various sub channels is achieved through the process ofmaximizing the channel capacity while minimizing the baud error rate.There exist in the literature a variety of optimal loading algorithmsthrough which the required energy distribution may be accomplished. Agood example may be found in patents: “Ensemble Modem Structure ForImperfect Transmission Media” U.S. Pat. Nos. 4,679,227, 4,732,826 and4,833,706.

With the aid of FIG. 5, an exemplary approach to optimized loading isnow discussed. Let the available channel bandwidth be divided into Nequal segments of length W. Assume that the frequency response withinthe i^(th) segment is flat and given by H_(i)(ƒ). Let the noise beadditive white Gaussian with double-sided spectral density N_(o)/2watts/Hz. Let the available signal power P be equally divided among allsub channels available, and normalize the system to the first subchannel. The received power in the i^(th) sub channel is thusP_(i)=l_(i)P/N where l_(i)=|H_(i)(ƒ)|²/|H_(i)(ƒ)|². It can then be shownthat a possible optimal choice of bit loading n_(i) is given by:

$\begin{matrix}{n_{i} = {\log_{2}\left\{ \frac{\left( {l_{i}/N} \right)\left( {3/2} \right)\left( {{P/N_{o}}W} \right)}{{- \ln}\;{\Pr(ɛ)}} \right\}}} & (19)\end{matrix}$

Where n_(i) is the number of bits allocated to the i^(th) sub channel,and Pr (ε) is the probability of symbol error for all sub channels. Inequation (19), since l_(i)P/N_(o)W is the signal to noise ratio in thei^(th) sub channel, a preferred embodiment of the invention will use ameasured value of the noise in the i^(th) sub channel for thecomputation of n_(i). This combined approach to the allocation ofsignaling energy and of bits to each sub channel comprises a specificoptimal approach to water filling.

A block diagram of the complete transmitter/receiver pair is shown inFIG. 2. In FIG. 2 the transmitter comprises N sub-transmitters 201-1 to201-N and the summer 202. Input data for transmission through thechannel are modulated at each sub-transmitter 201-1 to 201-N, and theoutputs are summed at the summer 202 for transmission through thechannel characterized by the function block 203. Addition of noise intothe system is depicted by function block 204 in FIG. 2. The i^(th)sub-transmitter 201-i is optimized in accord with water filling asdescribed above for the i^(th) segment of the channel. Similarly, thereceiver is comprised of N subcomponents 205-1 to 205-N, the i^(th)subcomponent 205-i corresponding the component 201-i of the transmitter.

The invention is now further described with greater specificity with theuse of FIG. 3, which illustrates how the discrete prolate functions areused for capacity optimization. With reference to function block 201-1,assume that, with the use of equation (19), a computed value of 6 wasobtained for n₁. It is clear from the foregoing discussion that thisloading bound can be assured through the resolution of the transmittedsignal with the use of two discrete prolate functions. Let n₁=n₁₁+n₁₂with n₁₁=2 and n₁₂=4. It can further be shown that, given the specificchoices for n₁₁ and n₁₂, if it is assumed that the symbol error is equalin both signaling dimensions, the power must be divided such thatP₁=P₁₁+P₁₂, where P₁₁=P₁/3 and P₁₂=2P₁₁. Given the foregoing choices ofparameters, an exemplary embodiment of the invention in function block201-1 is illustrated in FIG. 3. As can be seen from the figure, the sixbits to be transmitted are segmented at function block 301 into 2- and4-bits packets that are sent to function blocks 302-1 and 302-2. Atfunction block 303-1, a 4-level I/Q mapper is used, while a 16-levelmapper is used at function block 303-2. Within function block 304-1 theI and Q components from function block 303-1 and the power P₁₁ are usedto generate the in-phase and quadrature components of the prolate pulsescorresponding to p₁(t). Further, these components are modulated atfunction block 306-1 and 308-1, then summed at function block 309 foroutput to the channel. As can be seen from FIG. 3, similar activitiesoccur for the dimension corresponding to p₂(t)).

The structure of the optimized sub-receiver 205-1, associated withsub-transmitter 201-1, is shown in FIG. 6. As discussed earlier, the keyto retrieving the bits that were sent in a particular dimension is theuse of a low pass eigenfilter for that dimension. The discrete prolatepulses are thus used to form a low pass orthogonal filter bank forextracting the bit information from each dimension. The demodulatedI_(k) and Q_(k) values finally go through a reverse mapping process,after which the original block of bits is reconstructed.

In the receiver of FIG. 6 the channel output is received as indicated bythe block 601. This channel output is connected to a plurality of mixers603-1, 603-2, 605-1 and 605-2 and are mixed with cosine and sinesignals, respectively. These mixed signals are demodulated in theorthogonal filter bank containing filters 606-1, 606-2, 606-3 and 606-4.I/Q reverse mappings are performed in reverse mappers 607 and 608 torecover the segmented bits and the originally transmitted bit pattern isreconstructed in block 609. While discrete blocks are illustrated, theprocesses are stored program processes that are performed independentlyof block identification.

Synchronousness being of critical significance to the design oftelecommunication systems, reference is now made to the fact that in theconstruction of FIG. 2, FIG. 3, and FIG. 6, this requirement isstipulated. Thus, in a complete embodiment of the present invention,methods of carrier tracking and symbol recovery shall be implemented.There are various procedures well documented in the literature toaccomplish these operations. One reference describing synchronism withrespect to carrier tracking and symbol recovery is the text “DigitalCommunications, Fundamentals and Applications” by Bernard Sklar.Information specifically related to synchronization may be found inchapter 8, Pages 429-474.

Recall that in equation (2) ε was used to determine the pulseefficiency. Thus, in a preferred embodiment equation (9) may be used toshorten the length of the filtering process, in an effort to seekimplementation efficiency. Filtering must then be normalized by a factorof I/λ_(i) for each finger. In this case, keeping jitter to a minimumwill be a critical issue.

In present-day communication systems, because of the inefficiencies thatoccur with the application of a single signal for information bearing,the implementation of complex equalization structures is imperative toachieve the most efficient use of the channel. With the implementationof the design discussed herein, the equalizer shall effectively bereduced to a simple scaling function.

The invention presented herein was described in light of a preferredembodiment. It should be understood that such preferred embodiment doesnot limit the application of the present invention. Persons skilled inthe art will undoubtedly be able to anticipate alternatives that aredeemed to fall within the scope and spirit of the present application.

1. A method for receiving a multi-dimensional high-speed signalcomprised of a plurality of bits, via a modem, comprising steps of:receiving a channel output having a plurality of bits where each bit issent in a particular dimension; mixing signals of the received channeloutput with trigonometric signals; and retrieving bits in each of theparticular dimensions by: demodulating the mixed signals in anorthogonal filter bank of low pass Eigen filters defined by atransmitted prolate pulse of the channel output; reverse mapping thedemodulated mixed signals to recover segmented bits; and reconstructingan originally transmitted bit pattern to recreate an original signal. 2.The method of claim 1, wherein at least one of said mixed signalscomprise I_(k), and Q_(k) values, wherein the step of demodulatingdemodulates at least one of said mixed signals and wherein the step ofreverse mapping reverse maps the demodulated mixed signals in parallelpaths to reconstruct an original transmitted block of bits at atermination of the parallel paths.
 3. A high-speed receiver forreceiving a multi-dimensional high-speed signal comprised of a pluralityof bits, via a modem, comprising: an input for receiving a channeloutput signal having a spectral distribution defined by a prolate pulse;a mixing circuitry, connected to said input, to apply trigonometricmixing signals to the channel output signal; a demodulation circuitry,connected to said mixing circuitry, including a plurality of orthogonalfilter banks of low pass eigen filters defined by a transmitted prolatepulse of the channel output; and a reverse mapping circuitry connectedto said demodulation circuitry for reconstructing an originallytransmitted bit pattern to recreate an original signal.
 4. A method ofhigh-speed communications for achieving high spectral efficiencycomprising steps of: dividing a transmission channel into a plurality ofsub channels; modulating signals in each sub channel in a numberedplurality of sub transmitters using discrete prolate functions forcapacity optimization; summing the modulated signals and transmittingthe summed signals through a transmission channel subject to noise;recovering the individual channels in a plurality of receiverscorresponding to the plurality of sub channels; and reconstructingoriginal bits of the transmission channel by at least one demodulationtechnique.
 5. The method of claim 4, wherein the step of reconstructingthe original bits includes filtering in low pass Eigen filters.
 6. Themethod of claim 5, further comprising the step of: reverse mapping oneor more demodulated signals from said at least one demodulationtechnique to recover segmented bits of an originally transmitted bitpattern of a high-speed input signal.